Nomogram for determining shield thickness for point and line sources of gamma rays

N U C L E A R S T R U C T U R A L ENGINEERING 1 (1965) 324-331. N O R T H - H O L L A N D PUBLISHING COMP., A M S T E R D A M

NOMOGRAM FOR DETERMINING POINT AND LINE SOURCES

SHIELD THICKNESS FOR OF GAMMA RAYS *

C. J O N E M A L M

Aktiebolaget Atomenergi, Studsvik, Nyk~ping, Sweden Received 22 January 1965

A set of n o m o g r a m s is given for the determination of the r e q u i r e d shield thickness against gamma radiation. The s o u r c e s handled are point and infinite line s o u r c e s with shields of Pb, Fe, magnetite concrete ( 0 = 3 . 6 ) , ordinary c o n c r e t e ~0 = 2.3) or water. The gamma energy range covered is 0.5 10 MeV. The n o m o g r a m s are directly applicable for source and dose points on the s u r f a c e s of the shield. They can easily be extended to source and dose points in other positions by applying a g e o m e t r i c a l correction.

I. I N T R O D U C T I O N The most c o m m o n problem in shielding calculations if the estimation of the shield thickness when source strength, shield material and allowed dose rate are known. There exist nomograms for estimation of dose rate when source strength, shield thickness and material are known (see e.g. ref. [1]). However, n o m o g r a m s of this type are not easily applied in the reverse case when the shield thickness is wanted. In order to facilitate the determination of shield thicknesses for point and line sources a new type of n o m o g r a m has been prepared in which the previously mentioned drawbacks have been avoided. This set covers g a m m a energies between 0 5 and 10 M e V and the following materials: Pb, Fe, magnetite concrete (p = 3.6 g/cm3), ordinary concrete (p = 2.3 g / c m 3) and water. The user is only required to know source strength, g a m m a energy, shield material, the dose rate allowed and in s o m e cases the escape probability for g a m m a quanta from the source. The use of the n o m o g r a m s can be extended even to volume sources, as these in most cases can be approximated by one or m o r e point or line sources. In the n o m o g r a m s the escape probability is unity, which m a y be considered true in m a n y practical problems. If it is less than unity the source should of course be reduced by this factor. It would have been desirable to m a k e the n o m o g r a m s valid for an arbitrary distance be-

R ShieLd

Line source

point

,=

x

Fig. 1. Shielded line source geometry. t w e e n s o u r c e a n d d o s e p o i n t . To a v o i d m a k i n g the nomograms too complicated this distance is t a k e n e q u a l t o t h e t h i c k n e s s of t h e s h i e l d (R = x , f i g . 1). H o w e v e r , t h e n o m o g r a m s c a n a l s o b e u s e d when R > x by applying a g e o m e t r i c a l c o r rection described later.

2. S O U R C E S

OF DATA

For preparing the n o m o g r a m s the usual methods for calculating dose rate from point and line sources have been used [2]. Thus for a point source we have used

So B(px) exp(-~x) k D =

* Accepted by J. Braun.

(I)

4~R 2

NOMOGRAM FOR DETERMINING SHIELD THICKNESS and for a line s o u r c e

A1F[ ~'#;Ux(I +al )] +A2F[ ½~;px(I +~I )] D = SLk 2~R

(2) where D = dose r a t e in m r / h , SO = s o u r c e s t r e n g t h in M e V / s (point s o u r c e ) , SL = s o u r c e s t r e n g t h in M e V / s . c m (line source), ~ = total g a m m a - r a y a t t e n u a t i o n coefficient in c m - 1 , x = shield t h i c k n e s s in c m , B(px) = dose buildup f a c t o r , A1, A2, a l , a 2 = T a y l o r dose buildup coefficients, k = c o n v e r s i o n factor f r o m g a m m a flux to dose r a t e [3], and R = d i s t a n c e between dose point and source. In these n o m o g r a m s R = x. The values of p are those given by Grodstein [4], reproduced for instance in A N L - 5 8 0 0 [5]. The values for magnetite concrete are the ones calculated by Roos [6] based on p values given by Strom et al. [7]. F o r point s o u r c e s the buildup f a c t o r s u s e d a r e those given by Goldstein and W i l k i n s [8]. F o r c o n c r e t e s the f a c t o r s for A1 have been used. It should be pointed out, however, that t h e r e e x i s t s l a t e r c o m p i l a t i o n s of f a c t o r s for c o n c r e t e , for i n s t a n c e , those by Walker and G r o t e n h u i s [9] and by HSnig [10]. In the c a s e of o r d i n a r y c o n c r e t e , the r e s u l t s for A l - b u i l d u p a r e slightly i n a c c u r a t e but at l e a s t for the higher g a m m a e n e r g i e s c o n s e r v a t i v e . In m a g n e t i t e c o n c r e t e the A1 buildup gives a v e r y good a g r e e m e n t with that obtained u s i n g f a c t o r s by Walker and Grotenhuis. F o r the line s o u r c e s coefficients a r e t a k e n f r o m the o r i ginal work by T a y l o r [11], even t h e s e b a s e d on data by Goldstein and W i l k i n s [8]. The T a y l o r coefficients have also been c a l c u l a t e d by B u s c a g lione and M a n z i n i [12], whose work gives a fit to G o l d s t e i n - W i l k i n s v a l u e s which is even b e t t e r than that of the values by T a y l o r [11]. The e r r o r in the buildup c a l c u l a t e d by v a r i o u s T a y l o r coefficients is u s u a l l y r a t h e r s m a l l , even in t h e w o r s t case l e s s than 20%. F o r line s o u r c e s and m a g n e t i t e c o n c r e t e shields T a y l o r coeffic i e n t s for A1 have also b e e n used. They give a good a g r e e m e n t with r e s u l t s produced u s i n g the v a l u e s of W a l k e r and G r o t e n h u i s [9].

3. THE USE O F THE NOMOGRAMS The p r o c e d u r e for the n o m o g r a m s a p p e a r s in fig. 4. If the d i s t a n c e R between the s o u r c e and dose point i s g r e a t e r than the shield t h i c k n e s s x (fig. 1) a c o r r e c t i o n for the g e o m e t r i c a l a t t e n u a tion must be made. In t h i s c a s e a f i r s t value x 1 i s r e a d , a s s u m i n g R = x 1. T h e n a new value of x 2

32S

is r e a d with the s o u r c e s t r e n g t h r e d u c e d by the factor (×l/R)2 in the c a s e of a point s o u r c e and by (Xl/R) in the c a s e of a line source. This p r o c e d u r e should be r e p e a t e d once m o r e . If the change in x between the last two v a l u e s is s m a l l enough, the last value of x is the r e q u i r e d r e sult. If not, the i t e r a t i o n should be r e p e a t e d twice, b e c a u s e the total n u m b e r of i t e r a t i o n s must always be odd in o r d e r to get r e s u l t s on the safe side.

4. E X A M P L E We have a g a m m a s o u r c e of 1011 M e V / s with the g a m m a e n e r g y 1.5 MeV. The g e o m e t r i c a l d i m e n s i o n is so s m a l l that it can be c o n s i d e r e d a point s o u r c e . The escape p r o b a b i l i t y for the s o u r c e is 0.5 and it is situated 40 c m f r o m the i n n e r s u r f a c e of the shield. Design and i r o n shield for a dose r a t e of 1 m r / h . To b e g i n with the source strength is r e d u c e d by the escape p r o b a b i l i t y giving 5 × 1010 M e V / s and a shield t h i c k n e s s of 31 c m (fig. 4, dotted line). The g e o m e t r i c a l factor J(R) 2 =(\4040+31/~2 = 0.32. Reducing s o u r c e s t r e n g t h by t h i s factor we get 1.6 x 1010 M e V / s and a shield t h i c k n e s s of 27 cm. Now the new g e o m e t r i c a l factor is

40 ~2

40+27/

=0.36, and the r e d u c e d s o u r c e s t r e n g t h

is 1.8 x 1010 M e V / s . The final shield t h i c k n e s s is 28 cm.

5. HANDLING OF FINITE LINE SOURCES As the n o m o g r a m s a r e valid for infinite line s o u r c e s a lower dose r a t e than specified is obt a i n e d for a s o u r c e of l i m i t e d length ( 0 < ½ ~ , fig. 1). Table 1 gives the deviation obtained by r e p l a c i n g a finite line s o u r c e (O = 10o - 60 o) by an infinite line s o u r c e (0 = ½~). The deviation may evidently be neglected. If the line s o u r c e is short the spatial v a r i a t i o n of the flux will c l o s e l y a p p r o x i m a t e that of a point source. Thus for s m a l l O the line s o u r c e m a y be r e p l a c e d by a point source at the c e n t e r of the line having a s o u r c e s t r e n g t h S'O=SLh (fig. 1) The e r r o r s involved in this a p p r o x i m a tion a r e indicated in t a b l e 2 [2]. As we see in these t a b l e s it can be r e c o m mended that in n o r m a l c a s e s (px >t 5) a line s o u r c e with 0 > 20 ° be r e p l a c e d by a point s o u r c e and a line s o u r c e with 0 >i 20 ° by an infinite line, but to be s u r e the u s e r should always consult the

326

C. J(~NEMA LM REFERENCES

Table 1 D(O = l y ) / D ( o )

10 20 30 40 60

lax = 1

IAx= 5

p~ = 11

lax = 25

5.16 2.60 1.76 1.38 1.07

3.05 1.60 1.22 1.06 1.00

2.16 1.34 1.06 1.00 1.00

1.15 1.00 1.00 1.00 1.00

Table 2 D(point source)/D(line soureel $2x=0 1 5 10 20 30

1.00 1.00 1.01 1.04 1.10

~= 1.00 1.00 1.02 1.07 1.16

1

/2x= 5

lax= 11

1.00 1.00 1.04 1.15 1.38

1.00 1.02 1.07 1.32 1.70

p x = 25 1.00 1.04 1.14 1.55 2.37

t a b l e s to get the b e s t a p p r o x i m a t i o n . In f i g s . 2 - 1 1 t h e n o m o g r a m s a r e g i v e n f o r p o i n t and l i n e s o u r c e s f o r t h e m a t e r i a l s m e n t i o n e d , P b in f i g s . 2 - 3 , F e in f i g s . 4 - 5 , m a g n e t i t e c o n c r e t e i n f i g s . 6 - 7 , o r d i n a r y c o n c r e t e in f i g s . 8-9 a n d w a t e r i n f i g s . 10-11.

[1] S.E. Pihlajavaara and E. Pihlman, Gamma attenuation nomogram for concrete, The State Institute for Technical R e s e a r c h Publication Series III, No. 52 (Helsinki, 1961). [2] T. Rockwell III, Reactor shielding design manual. [3] H. Goldstein, Fundamental a s p e c t s of r e a c t o r shielding (Pergamon P r e s s , London, 1959). [4] G. W. Grodstein, X - r a y attenuation coefficients from 10 keV to 100 MeV, Report NBS-C-583 (1957); R. T. Ginn~es, X - r a y attenuation coefficients from 10 keV to 100 MeV, Suppl. to Report NBS-C-583 (1959), National Bureau of Standards, Washington. [5] Reactor physics constants, ANL-5800. [6] M. Roos, KArnfysikaliska egenskaper hos R 3 / Adams tunga betong, RSA-17/R3-135 (1959). [7] E. Strom, E. Gilbert and H. Israel, G a m m a - r a y absorption coefficients for e l e m e n t s 1 through 100 derived from the theoretical values of NBS, LA2237 (1959). [8] H. Goldstein and J . E . W i l k i n s , Calculation of the penetration of gamma rays, NYO-3075 (1954). [9] R. L. Walker and M. Grotenhuis, A summary of Shielding constants for concrete, ANL-6443 (1961). [10] A. Hbnig, Dosiszuwachsfaktoren bei Beton, K e r n technik 9 (1964). [11] J.J. Taylor, Application of gamma ray buildup data to shield design, WAPB RH-217. [12] S. Buscaglione and R. Manzini, Buildup f a c t o r s coefficients of the J . J . Taylor equation, t r a n s l a t e d from the Italian in ORNL-TR-80.

NOMOGRAM FOR DETERMINING SHIELD THICKNESS

327

Source: point ir,otropic

~.xN ?.,. \

_

.'x

,

~w

N

t.{l~i¢,

\

"

\

~

Fig. 2. I s o t r o p i c point s o u r c e - lead shield. [,

,

5

10

15

20

25

30

35

40

l?ig. 3. I n f i n i t e ].ine s o u r c e - l e a d shie~.d.

45

)50pot:m) 55

CO

328

C. J O N E M A L M

~ 10, lO5 lO6 ~ , ~

~ lo. 10, ~

\

~ 10~

101.

I Mat~iat: Fe

\~\ \~ 0

10

20

30

60

70

80

90

- Fe

Fig. 4. I s o t r o p i c p o i n t s o u r c e - i r o n s h i e l d .

s(M,VI~s) ~ I/ I0' ~ lo' 10' io' Io' f f io" 10~10" 10'<10'~10" 10' Io"

\

~~

10 F i g . 5. I n f i n i t e l i n e s o u r c e - i r o n s h i e l d .

Ill

'

NOMOGRAM FOR DETERMINING SHIELD THICKNESS

III

0

20

Fig. 6. Isotropie point source - magnetite

•

concrete

shield.

) = 3.6 9 / c r ~

F i g . ~. I n f i n i t e l i l l e ~ o u r e e - m a g n e t i t e c o n c r e t e s h i e l d .

3 ~'9

c . JC)NEMALM

330

I (p = 23 g/crr?)

\ 0

30

60

90

120

150

180

210

~

2/0

Fig. 8. Isotropic point source - o r d i n a r y c o n c r e t e shield.

s(~V/~s)

~o~ ~

# ~s # d ~ ~ ~ ~" ~12~0. #

•

~o' ~

~o~ ~o"

~= 2.~ ~ )

\ \ \

o

30

6o

90

~o

~o

\\:

~o

~o

=o

~

=~.aoL~ j p

Fig. 9. Infinite line s o u r c e - o r d i n a r y concrete shield.

~o

NOMOGRAM FOR D E T E R M I N I N G S H I E L D THICKNESS

_~.s(~V/.)

100

150

331

~

e-~lO,~o,~, ~ ~,~o.~o~o~o~o,~.d, ~,

200

250

300

350

400

450

Fig. 10. I s o t r o p i c p o i n t s o u r c e - w a t e r s h i e l d .

.

\ 0

50

100

150

200

250

\\\\ 300

350

\ 40(}

Fig. 11. I n f i n i t e l i n e source - w a t e r shield.

450

5~0 ~

550 (cm} 600